### 6.1 Characterization of the channel realizations

In order to ensure statistically representative results, we conducted a sufficiently large number of executions of the aforementioned procedure over different wireless channels. In particular, binary switches allowed us to choose four different two-antenna sets at each node which makes a total of 4096 different channel realizations (all channels are available for download in the web page of the COMONSENS project [30]). The estimated RMS delay spread for all channel realizations is 180.7 ns, which is a relatively long delay spread but in accordance with values reported by the TGn channel models [33]. Note that the corresponding interference alignment solution is completely different as long as a single channel coefficient of the three channel matrices changes. We recall that the position of the nodes nor the transmission frequency were changed.

First of all, we characterized the quality of the channels in our setup. In Fig. 10, we provide an example of the frequency response magnitude (normalized by the average of the channel amplitudes) for one of the measured 2×2 MIMO channels. Figure 10 also plots the estimated noise power at each receive antenna obtained as indicated in Table 1. Notice that we can obtain four noise variance estimates, one for each transmit-receive antenna pair. It can be observed that the noise level is not flat over frequency and follows the quality of the corresponding channel coefficient, i.e., it is proportional to the channel gain. This behavior is explained by signal distortion at the transmitter, also referred to as transmitter noise [34]. Hereinafter, we will refer to this estimated noise as residual noise according to the way it is calculated.

Then, we define the signal-to-residual noise ratio (SRNR) as the ratio between the estimated signal power and the residual noise power. Notice that the SRNR serves as a pessimistic proxy for the SNR as it accounts for the combined effect of the thermal noise at the receiver and the signal distortion at the transmitter. Figure 11 shows the estimated probability density function (PDF) of the SRNR at each receiver for both the desired and the interfering links. The SRNR for each subcarrier has been obtained with the expressions indicated in Table 1. As shown in the figure, the SRNRs range from approximately 15 to 30 dB, with significant differences among receivers: interference is slightly stronger than signal at receivers 1 and 3, whereas receiver 2 experiences higher signal strength.

These measurements demonstrate the suitability of the scenario for the evaluation of IA techniques: first, all desired and interfering signals are of comparable strength and, second, SRNRs are relatively high.

### 6.2 Comparison of pre-FFT and post-FFT IA decoding

In this section, we experimentally evaluate the performance of the pre-FFT (time-domain) IA decoding scheme proposed in Section 2.2 in comparison to post-FFT (frequency-domain) decoding while at the transmitter, IA precoding is applied in the frequency domain and the three transmitters are synchronized among them. All OTA transmissions are carried out at 24 Mbit/s (16-QAM) and the EVM of the received signal constellation (calculated as in Table 1) is used as the performance metric.

We start assessing the performance of both decoding schemes in a scenario where users transmit with controlled time delays between them. Notice that, although all measurements have been carried out under synchronized transmissions, we can arbitrarily control the time delay between the transmitters by using the received signals during the perfect IA transmission stage. Indeed, in the perfect IA transmission stage, each receiver node acquires an interference-free version of the signal transmitted by each user. After synchronizing those frames and before continuing their processing, an arbitrary delay can be included in each of the received signals, which are then summed up to yield a received signal comprised of the three time-misaligned frames. In order to ensure that the desired signals are entirely affected by interference, we apply such a time delay as a cyclical shift, increasing the total length of the received signal accordingly. Note that the resulting SRNR will be approximately 4.7 dB lower, since the noise is also added up in this process. Nevertheless, the relative performance of pre-FFT and post-FFT IA decoding will not be noticeably affected by such SRNR reduction. We plot in Fig. 12 the estimated median EVM of the received constellation for *M*=30 training symbols and a decoder length of *L*=30 samples. It can be observed that the EVM of the post-FFT decoding scheme degrades as the delay increases, which is the result of the aforementioned synchronization issues inherent to the post-FFT approach. On the contrary, the pre-FFT decoding scheme exhibits an EVM almost independent from the time delay. Finally, the third curve labeled as “sync-aided IA post-FFT decoding” shows the EVM of the post-FFT decoding scheme when the time synchronization is perfect (no time synchronization tasks are performed, since the optimum delay is known beforehand it is directly applied at the receiver), and the resulting EVM is similar to that of the pre-FFT scheme.

To further evaluate the pre-FFT IA decoding scheme, we will focus on the synchronized setting in the remainder of this section. We first study the impact of the pre-FFT decoder length on the performance of IA which, as mentioned in Section 2.2, involves a trade-off between ISI and residual MUI. To this end, we evaluate the EVM of the received signal constellation when MUI is suppressed with both post- and pre-FFT decoders. Training frames consist of *M*=30 training OFDM symbols per transmit antenna. Figure 13 shows the median EVM degradation of the pre-FFT technique for different decoder lengths, *L*∈ [ 1,64], with respect to the post-FFT decoder which obviously provides the best performance. In order to demonstrate the ISI versus residual MUI trade-off, the comparison has been carried out for both IA and perfect IA transmissions. For perfect IA, the degradation is only due to ISI and, as expected, it increases with the decoder length. On the other hand, a shortened IA decoder cannot properly suppress the MUI leading to a high degradation of the constellation EVM. As the decoder length increases, however, the amount of MUI is greatly reduced whereas the degradation due to ISI grows at the rate seen in the perfect IA curve. This analysis illustrates the existing ISI-MUI trade-off from which it turns out that a good choice for the decoder length would be 30 taps. This decoder length will be used in the remaining experiments since it provides slightly less than 1 dB of EVM degradation (whereof around 0.3 dB are due to ISI) with the advantage of a reduced receiver complexity and the possibility to perform frame synchronization in totally unsynchronized scenarios (as revealed in Fig. 12).

Secondly, we evaluate the effect that the quality of the CSI has on the performance of aligned transmissions considering our setup. In Fig. 14, we show the evolution of the EVM for different numbers of OFDM training symbols. From the two upper curves (corresponding to IA transmissions), it can be observed that a small number of training symbols, below 20 or 30, does not provide an accurate CSI and leads to a significant degradation of the EVM due to interference. On the other hand, a number of training symbols above 30 does not improve the EVM anymore, which leads to a constant degradation between perfect IA and IA of around 4 dB. The fact that the EVM does not improve when increasing the number of training symbols suggests that the performance of IA is not only limited by imperfect CSI but also by other spurious effects, such as those derived from reusing the same training sequence and thus exciting the same nonlinearities each time. Notice also that the gap between post-FFT and pre-FFT decoding is substantially higher for IA than for perfect IA. This is due to the fact that, for the latter case, the degradation between both decoding schemes is caused by the additional ISI introduced by the pre-FFT decoding process (since the MUI has been avoided by sequential transmissions), whereas for the former case is due to ISI and residual MUI (see Fig. 13). Additionally, the impact of transmitter noise is also lower for perfect IA since a single user is transmitting at a time instead of three simultaneously, as in the case of pre-FFT or post-FFT IA.

One possible source of degradation could be the channel variations between the training stage and the data transmission stage, since the channel estimates used to compute the IA precoders and decoders are outdated by the time the aligned precoded transmission is actually performed. To evaluate this hypothesis, we conducted an additional experiment where a deliberate feedback time was introduced.^{2} The results in Fig. 15 show that increasing feedback time does not cause additional degradation of the received signal EVM, hence proving the channel remains static for at least 10 s. Notice, however, that the performance could improve for feedback times shorter than a second, which we cannot measure. This is consistent with the special care taken to guarantee that our measurement scenario is completely static (see Section 4). From the results shown in Fig. 15, we can ensure the validity of our measurement methodology regardless of the feedback time.

Once both the hypotheses of having inaccurate and outdated CSI estimates have been ruled out, there are still other reasonable effects which may jointly limit the performance of IA and may not completely disappear when using a large number of pilot symbols.

The impact of non-linearities in power amplifiers and RF oscillator phase noise on IA was empirically evaluated in [7]. When the signal distortion occurs at the transmitter, it is known as a transmitter noise and leads to spatially colored noise at the receiver. Transmitter noise, also referred to as dirty RF, is specially important when the transmitter and the receiver are close to each other since its effect is directly proportional to the channel power gain. Section 6.1 shows that transmitter noise is present in our measurement campaign. Its detrimental impact on the performance of MIMO systems is already well-known and has been empirically studied in [34–36]. Some other effects, such as amplification gain drift (also known as transmitter droop) [37] and packet-to-packet power variations, have not been studied yet in the context of IA. Notice that these effects may be specially pernicious for spatial-domain IA since they lead to power fluctuations at the transmitter over time which are different for every antenna and packet and therefore cannot be fought with training.

A completely different explanation for the degradation stems from the fact of applying precoding in the time domain at the transmitter side, i.e., on a per-subcarrier basis. As we pointed out in Section 6.2, the only way to suppress the interference independently of the delays between transmitters and receivers is to design both precoders and decoders to be applied in the time domain. In our experiments, as precoding is applied in the frequency domain at the transmitters, each receiver sees a residual interference that is proportional to the relative delay between the incoming desired and interfering signals. However, a careful experimental analysis of this issue and how largely it affects IA is still necessary and we leave it as a future work.

Finally, in view of the results in Figs. 13 and 14, we have chosen the parameters which provide a nearly optimal performance with a reasonable complexity, that is, *M*=30 training symbols and a decoder length of *L*=30 samples. The cumulative distribution function (CDF) of the received constellation EVM obtained with this parameter setup is shown in Fig. 16. It is shown that the performance loss caused by moving from a frequency-domain to a time-domain decoder is always below 1 dB for IA and below 0.5 dB for perfect IA (while, in both cases, IA precoders are applied in the frequency domain at the transmitter). As a counterpart, time-domain IA decoding has the advantage that no inter-user time synchronization is required. Additionally, these differences are negligible compared to the roughly 4-dB difference between perfect IA and IA schemes shown in Fig. 16.

Alternatively, we show BER results for both approaches in Fig. 17. This figure represents the average achievable sum-rate that guarantees a BER equal to or lower than a given value. For each channel realization, the achievable sum-rate is obtained assuming an optimal MAC layer which selects for each user the maximum rate that satisfies the required BER. It is important to notice that results in Fig. 17 do not take additional overhead or higher-level issues into account and they only suggest how the optimum performance of such schemes would be. The difference of 1 dB in terms of EVM between time-domain (pre-FFT) and frequency-domain (post-FFT) IA decoding (shown in Figs. 13, 14, and 15) translates into a noticeably higher gap in terms of sum-rate, as shown in Fig. 17. Note also that more sophisticated algorithms may help reduce the gap between pre-FFT and post-FFT, but this analysis is out of the scope of the paper and we leave it as future work.

### 6.3 Comparison of the adopted schemes

In this subsection, we compare the performance of the five adopted schemes using two different metrics. With respect to IA, in this subsection, we only consider post-FFT (frequency-domain) IA decoding. First, we show in Fig. 18 the CDF of the received signal constellation EVM. As expected, DET-TDMA provides the lowest EVM and guarantees an EVM better than −15 dB for all channel realizations, whereas IA ensures the same signal quality in 60 % of the realizations. On the other hand, Fig. 18 also shows a noticeable degradation of IA with respect to perfect IA, where the latter is able to achieve the same EVM value of −15 dB in a 20 % more of channel realizations. This effect was already observed in Fig. 16 and is due not only to channel estimation errors, which avoid the interference to be perfectly nulled out, but also to transmitter noise and synchronization issues, as already explained in Sections 6.1 and 6.2. Alternatively, the MaxSINR scheme provides little EVM improvement over IA, increasing the percentage in only 4 % at −15 dB. This suggests that the operating SNRs are sufficiently high for IA to achieve good performance, and therefore MaxSINR algorithm converges to the zero-forcing IA solution in most subcarriers. However, when there exists high collinearity between the signal and the interference subspaces, MaxSINR enhances the desired channel, thus providing an improvement in the average EVM performance. Finally, it is worth mentioning that the quality of the equivalent channels after applying the IA precoders and decoders, which is represented by the EVM performance of perfect IA, is more spread than that of SISO channels. This is a reasonable result since IA precoders and decoders are independent of the desired links, hence yielding collinearity as well as orthogonality between the signal and the interference subspaces with the same probability.

Finally, Fig. 19 shows the BER results for the five adopted schemes. We observe that IA schemes achieve higher throughput than TDMA schemes for all BER requirements. For instance, IA provides an average rate of 73 Mbit/s with a maximum BER of 10^{−4}, whereas SISO and DET achieve 32 and 53 Mbit/s, respectively. On the other hand, although MaxSINR does not provide a significant improvement in terms of EVM (see Fig. 18), it does provide substantially higher data rates than IA. More specifically, it achieves 7 Mbit/s more than IA at the same operating point of BER≤10^{−4}. This is due to the fact that the channel encoding is sensitive to changes in the received EVM and thus a small improvement in the signal quality may yield a significant BER decrease, hence showing the importance of enhancing the signal quality when collinearity between the signal and the interference subspaces occurs. Following these lines, we also observe that perfect IA provides a large throughput improvement over IA, which evidences once again the significant impact of practical impairments such as channel estimation errors, transmitter noise, and imperfect timing. Such impairments, along with collinearity issues, significantly limit the performance of IA schemes (specially as the number of users increases) and should be considered in future theoretical IA designs.